So I have decided that picking up an ally to carry them around costs one attack to grapple, no roll required and no reduction of movement if the target is willing. Because the target is willingly being moved however, it still provokes an attack of opportunity.
The issue with allowing this maneuver as a free object interaction, or even as a bonus action, is that players will be able to abuse this in a thoroughly absurd way by playing leapfrog every round. Imagine two PCs are being chased by a guard. The guard Dashes 60 feet on its turn, then Player 1 picks up his buddy as a free action and Dashes 60 feet. Player 2 now picks up Player 1 and Dashes an additional 60 feet, effectively doubling their movement speed and making it impossible for the guard to catch up. Now make the PCs Rogues with Cunning Action... You see my point?
With my houserule, a Fighter can still make attacks after the carry provided they have Extra Attacks, but you have to take the Attack Action so you can’t Dash on the first turn carrying someone unless you have Cunning Action, Expeditious Retreat, or Action Surge.